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Analyzing Oil Refinery Investment Decisions:
A Game Theoretic Approach
ByFilip Ravinger
Submitted toCentral European UniversityDepartment of Economics
In partial fulfilment of the requirements for the degree of Master of Arts
Supervisor: Professor Andrzej Baniak
Budapest, Hungary2007
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Acknowledgement
I would like to thank my supervisor, Professor Andrzej Baniak, for his help and
suggestions that led to improvement and refinement of this thesis. I am also grateful
to two specialists from MOL Hungarian Oil and Gas Plc., Laszlo Varro and Peter Simon
Vargha for the insightful discussions and for providing data sources. Lastly, my thanks go
to my academic writing instructor, Reka Futasz, for her patience.
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Abstract
The objective of this thesis is to build a model of strategic behavior in an oligopolistic
market structure, which takes into account key features of the oil refinery industry, such
as heterogeneity of the output, large-scale investment with sunk costs and a high degree
of uncertainty over future payoffs. In view of recent demand shifts toward lighter refinery
products, the question arises whether it is profitable to undertake the upgrade investment
that enables to increase product yields of those products. Using game theory tools, multi-
product theory and a simple real-options analysis, a Cournot oligopoly model is devised
to attempt to answer that question and to assess the investment behavior of refineries.
It is argued that as long as some refinery enjoys a sufficient technological advantage, the
equilibrium will have only this refinery investing. Further, the model gives insight into
how changes in demand anticipations affect the equilibrium outcome of the investment
game. Finally, the applicability of the model is illustrated by a case study, focusing on
the Hungarian and Romanian oil refining sectors.
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Contents
Introduction 1
1 Modeling Oil Refinery Investment Decisions 4
1.1 Multi-Product Oligopoly Models . . . . . . . . . . . . . . . . . . . . . . . . 41.2 The Main Characteristics of the Refinery Sector . . . . . . . . . . . . . . . 51.3 The Basic Refinery Investment Model . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.2 Fixed Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.3 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.4 Upgrade Investment . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Solution of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.1 Fixed Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4.2 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4.3 Upgrade Investment . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Extensions of the Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . 141.5.1 Capacity Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.5.2 Multiple Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.6 Comparison with a Single-Product Case . . . . . . . . . . . . . . . . . . . 171.7 Qualitative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.7.1 The Input Choice Game . . . . . . . . . . . . . . . . . . . . . . . . 191.7.2 The Upgrade Investment Game . . . . . . . . . . . . . . . . . . . . 21
2 Modeling Refinery Investment under Uncertainty 24
2.1 Investment under Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . 242.2 Uncertainty in Refinery Investment Models . . . . . . . . . . . . . . . . . . 262.3 The Basic Model with Uncertainty . . . . . . . . . . . . . . . . . . . . . . 26
2.3.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3.2 The Case of One Refinery . . . . . . . . . . . . . . . . . . . . . . . 282.3.3 The Case of Two Refineries . . . . . . . . . . . . . . . . . . . . . . 29
2.4 Qualitative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4.1 One Refinery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4.2 Two Refineries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
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3 Application: Refinery Investment in Hungary and Romania 34
3.1 The Refining Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.1.1 Hungary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.1.2 Romania . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Demand Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.1 Hungary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.2 Romania . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Application of the Investment Model . . . . . . . . . . . . . . . . . . . . . 383.3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Conclusion 43
Appendix 46
A.1 Derivation of Optimal Input in the Basic Model . . . . . . . . . . . . . . . 46A.2 Derivation of the Slope of the Reaction Function . . . . . . . . . . . . . . . 47
References 48
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List of Figures
1.1 The cost and marginal cost functions . . . . . . . . . . . . . . . . . . . . . 111.2 Reaction curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.3 Reaction curve varying with ρ1,h . . . . . . . . . . . . . . . . . . . . . . . . 201.4 (a) Optimal input and (b) profit varying with ρ1,h . . . . . . . . . . . . . . 211.5 (a) Total cost and (b) marginal cost before and after investment . . . . . . 22
2.1 The tree of the game with one refinery . . . . . . . . . . . . . . . . . . . . 282.2 The tree of the game with two refineries . . . . . . . . . . . . . . . . . . . 30
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List of Tables
2.1 The expected payoffs in the two-refinery game . . . . . . . . . . . . . . . . 31
3.1 Calibration of the demand parameters . . . . . . . . . . . . . . . . . . . . 393.2 Parameter values before and after the investment . . . . . . . . . . . . . . 403.3 The matrix of optimal inputs . . . . . . . . . . . . . . . . . . . . . . . . . 413.4 The payoff matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.5 The payoff matrix of the two-period game . . . . . . . . . . . . . . . . . . 41
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Introduction
Since the emergence of the industrial revolution in the 18th century, the use of fossil
fuels has increasingly become a driving force of economic development. Since then, a
lot of economic research has been devoted to studying the use and management of non-
renewable natural resources. In particular, as petroleum is one of the most valuable
resources today, its efficient exploitation and processing has been a major field of interest
for a large number of researchers.
More specifically, an integral part of the petroleum value chain is refining, a process
that enables to deliver marketable oil output to the end consumers. From an economist’s
point of view, the oil refinery might be just an ordinary corporate entity, whose objective
is to maximize its value, subject to constraints on the supply as well as on the demand
side. Then, to examine the refinery’s behavior, she might apply some of the traditional
theories of the firm developed in economics. However, when doing so, the economist must
be aware of certain issues that are specific to the oil refining industry.
First and utmost, the output of a refinery is a composite product, and as such,
it is subject to competition on a variety of product markets. Second, the minimum
capacity of modern refineries is rather large relative to the demand in a regional market,
thus, adjusting the production process to meet the market demand involves large-scale
investment, which carries significant sunk costs. Third, entrepreneurship in oil refining is
subject to a high degree of uncertainty over future market conditions that can substantially
affect the value of refinery projects.
The objective of this thesis is to build a model of strategic behavior in an oligopolistic
market structure, which takes into account some key features of the refinery industry,
and to apply the model to an analysis of refinery investment decisions. In the short run,
refineries operate within given capacities and maximize their profit, given the market
conditions (e.g. the price of crude oil and of refined products). In the long run, the
refineries may alter their capacities based on their belief of future profitability (e.g. crude
price outlook, demand for and supply of refined products). Since there is a significant lag
between the investment decisions and their materialization, the decisions can be viewed
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as strategic commitments that carry considerable risk.
In view of the recent global demand shift from the low-value refined products (e.g. fuel
oil) to the high-value ones (e.g. gasoline, diesel), various ways of upgrading the refining
process to adapt this shift are fiercely debated. Apart from investing in higher capacities,
the refineries may have a possibility to invest in advanced technologies that enable them
to extract higher yields of the high-value products. However, both options are costly
and subject to uncertainty over potential future rewards. Hence, a proper analysis of the
investment motives must be undertaken, which is the main task of this thesis.
Due to a relatively high market share of each firm, the margin behavior of refineries is
probably best analyzed in the framework of an oligopoly model. In this thesis, a Cournot
oligopoly approach with production quantities as strategic variables is developed. If,
however, short-run decisions are made in strategic interaction among refineries, so are
long-run decisions. Thus, the investment decision can also be seen as a game. Hence,
this thesis aims to propose an oligopoly model, incorporating uncertainty and using game
theory tools to assess the investment behavior of the refineries.
The specific features of the refinery industry, the heterogeneity of output and
investment under uncertainty, are to be addressed. The former will be treated as
a special application of the multi-product oligopoly theory, while the latter can be
modeled as stochastic demand with different states of the world: high and low demand
situation for gasoline and diesel, the probabilities of which are common knowledge. These
particularities together may lead to interesting dynamics in investment behavior, e.g.
symmetric and asymmetric equilibria may exist. The model is to give some insight (in this
simplified framework), for example, into how the outcomes change if demand anticipations
are changed.
Some relevant literature that focuses on modeling the oil refining sector include the
following. Manne (1951), one of the first authors to extensively study the refining industry,
uses econometric techniques to estimate the cross-elasticities of the supply of the refined
products, based on which one can predict their relative prices. Later, he develops these
concepts to devise a linear programming model (Manne, 1958) and claims to answer
questions of the substitutability of different products. More recently, Pompermayer et al.
(2002) propose a spatial oligopoly model, which is close to the one treated in this thesis.
Using linear programming techniques, the authors study the behavior of refiners in the
Brazilian refinery market. Finally, Adhitya et al. (2006) use simulation techniques to
evaluate refinery supply chain policies and investment decisions.
While the literatures on multi-product theory and investment under uncertainty are
rich, treatment that would combine the two approaches is less so. Hence, through an
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application to the refining industry, one of the objectives of this study is to attempt to
fill this gap in the literature.
The remainder of the thesis is organized as follows. In Chapter 1, the basic
deterministic investment model is presented. Chapter 2 provides an extension by
introducing stochastics into the model. In Chapter 3, applicability of the model is
illustrated by a real-world case study.
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Chapter 1
Modeling Oil Refinery Investment
Decisions
The aim of the first chapter is to provide a detailed analysis of the basic deterministic
refinery investment model. We start with a brief introduction to multi-product theory and
a general description of the refinery industry. The investment model is then constructed
in steps, followed a qualitative discussion of the simple case of only two refineries.
1.1 Multi-Product Oligopoly Models
The study of oligopolistic market structures has been in the focus of the economic
literature since the publication of the seminal work of Cournot (1838). Oligopoly is
an industry form where a small number of firms dominate the market for a single
homogeneous product. With only a few producers in the market, it is reasonable to
expect that the actions of individual producers affect the overall state of the industry
and, in turn, the other producers’ performance. Game theory tools have been proved
very useful in analyzing such situations. In a setting where each firm pursues its own
benefit but must also consider the potential responses of its rivals, the central problem
is to find conditions which ensure the equilibrium state of the game. An intuitive notion
of an equilibrium, a state where none of the firms can gain by deviating from its current
strategy, is captured by what game theorists call the Nash equilibrium.
Many models have been proposed to analyze the behavior of firms in an oligopolistic
market structure. A distinguishing element of most of them is the way they view the
firms’ responses to each other and to the market. In the model of the oil refinery oligopoly
treated in this paper, we consider the production strategies to induce interaction among
refineries, which leads us to applying the model of the Cournot quantity competition. In
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1.2 The Main Characteristics of the Refinery Sector
this model the firms are assumed to compete in quantities, that is, their decision variables
are the amounts of output to be produced. The equilibrium concept to be used here is
that of a Cournot or Cournot-Nash equilibrium.
In reality, however, firms often produce more than one product, and thus competition
among producers involves interaction on more than one market. This important
generalization of the standard oligopoly models has been widely addressed in the literature
and is also in the focus of this paper. In a multiproduct oligopoly game the firms seek to
maximize their profits by choosing quantities of each product. If the supply of and the
demand for every product is independent of one another, this problem can be restated
in terms of single-product games, so that in every market a standard Cournot game is
played. Yet, this is seldom the case. On the supply side, the technology can be such
that it involves a joint production process, while on the demand side, the products can be
complements or substitutes. Then, more sophisticated methods of analysis are to be called
for. Still, an underlying objective of these methods is to find conditions that guarantee
the existence of the Cournot equilibrium. Okuguchi and Szidarovszky (1990) provide
an extensive treatise on multi-product oligopoly theory. Fortunately, as we will see, the
specificities of the refinery sector will allow us to devise a relatively simple multi-product
model, which will be under certain circumstances representable also in a single-product
form.
1.2 The Main Characteristics of the Refinery Sector
Crude oil is extracted from the ground in a form that is not suitable for market delivery.
It contains many impurities and contaminants, which need to be separated through a
refining process. This is a complex procedure involving a sequence of chemical processes,
such as distillation, catalysis and hydrotreating. It includes a flow of various intermediate
product streams, which yields as a final outcome a range of marketable end products
as gasoline, diesel and fuel oil. The complexity of the whole process varies with different
chemical properties of the oil input and so does the associated cost. In particular, the core
of the refining process lies in the separation of crude-oil ingredients with diverse weight -
hydrocarbons and other compounds such as sulfur, nitrogen and oxygen. It is primarily
the weight of these ingredients that determines the qualitative properties of the crude
(and hence its price), which in turn determine the yields of the refined end products and
their qualitative properties (and their price).1
Hence, the refinery is viewed as a multi-product firm that uses crude oil as input
1For more on the subject of the quality and pricing of crude oil, see Manes (1964).
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1.2 The Main Characteristics of the Refinery Sector
and produces a set of final products. It is important to realize that given the available
technology, the type of crude oil, the operating mode of the refinery and the complexity of
the refinery, the yields of the refined products are a result of a particular refining process
and so are technology-specific. This means that the production of each product depends
on the production of the other products. This special characteristics of the production
can be referred to as the inverse Leontief technology,2 where a certain amount of one
product cannot be produced without producing certain amounts of other products at the
same time.
Moreover, the demands for the refined products need not be independent. Specifically,
gasoline and diesel can be viewed as substitutes (although imperfect) by consumers. Also,
horizontal or vertical product differentiation by different refiners is likely to be an issue.
These points may well complicate the building of our refinery oligopoly model.
An obvious consequence of the described characteristics of the refinery sector is a
problem of a potential imbalance between the industry demand and supply. In fact,
over years it has been a major challenge for the refineries to match their product slate
with what the consumer wants. The demand for some products has been growing in
exchange for a decreasing demand for some other ones and, due to the rigid character
of the technology, the refinery supply has increasingly got out of balance with demand.
Particularly, the preference for lighter products, those more easily refined from lighter
oils, has become prevalent. For refiners it is the main challenge to adjust their refining
process to be able to meet the varying preferences for different products.
Although a complex task, the adjustment can be done by investing in facilities that
enable to chemically reprocess some intermediate or residual products to finally yield
lighter products. As a result, the refinery is able to extract more of the lighter, more
valuable products and to reduce the yields of heavier, less valuable ones. For example,
through the process of hydrocracking the refineries can increase their yields of diesel, while
fluid catalytic cracking enables them to raise gasoline yields. These processes, however,
involve costly investment in new cracking plants, so the refinery must carefully analyze
the profitability of their investment decisions. Moreover, since the minimum efficiency
scale in the refinery industry is rather large relative to the market, investment is of a
rather discrete nature.3 Consequently, this investment is seen as a strategic action and
must be analyzed in the context of the oligopolistic structure of the industry.
2Leontief technology assumes fixed proportions of input that produce a single output, while here wehave a single input that produces fixed proportions of output, hence the term ”inverse”.
3That is, it is hardly feasible for a refinery to invest in a relatively small hydrocracking unit and thento continuously adjust its capacity.
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1.3 The Basic Refinery Investment Model
The CERA4 Report by Kennaby (2003) provides an insight into recent developments
in the refinery sector in Europe. According to the report, the demand for European heavy
fuel oil has been steadily decreasing over the 1990s and is expected to continue to fall
further. On the other hand, the demand for middle distillates (e.g. diesel) has been on the
rise and is expected to continue increasing. This shift in the demand profile seems to favor
investing in hydrocracking processes, but the cost is a major barrier. Thus, many of the
simple refineries have even started to face a choice between investment and closure. For
the more complex refineries, the overall trend is to undertake investment in hydrocrackers,
but which refinery actually invests and which does not can be viewed as a game. We will
analyze this game by building a simple refinery investment model.
1.3 The Basic Refinery Investment Model
Our aim is to devise a multi-product model that would take into account the specific
features of the refinery production technology outlined above. The underlying interaction
of the players is modeled as a static non-cooperative game. The firms in the refinery
industry make production decisions in advance, so they can be viewed as competing in
quantities. Hence, the Cournot model appears to be convenient to be applied here.
We consider an elementary oligopoly model of oil refineries as follows. Although we
mentioned earlier that crude-oil types vary significantly in their physical properties, here
we assume that each refinery uses crude oil as a single homogeneous input. However, the
product line is not a single homogeneous good, rather, it consists of several dozen fuels and
chemicals. In particular, through a process of simple distillation the refineries successively
separate lighter products (e.g. liquid petroleum gas, naphtha, gasoline), middle distillates
(e.g. jet fuel, diesel, kerosene) and heaviest products (residual fuel oil). These products are
differentiated by their qualitative characteristics and desirability, which determine their
final value. Moreover, the products are refined in fixed proportions, that is, out of one
unit of crude oil input, the refinery can extract a specific share of each output, depending
on the current technology and the operating mode. It may then be in the interest of
refineries to invest in improved technologies, which increase the yield of the higher-valued
products. This is done through downstream processing, whereby the heavy feedstock is
reprocessed and changed into lighter, more valuable output. Consequently, the question
arises whether and to what extent it is profitable for the refineries to undertake this
upgrade investment, taking into consideration the oligopolistic structure of the refinery
industry and hence the strategic interaction among refineries.
4Cambridge Energy Research Associates.
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1.3 The Basic Refinery Investment Model
1.3.1 Setup
We start with a general setting of the model. We denote by R the number of refineries
operating on a particular market and by Z the number of refined products. Each refinery
r = 1, . . . , R acquires crude oil from the crude market, the amount of which is denoted by
xr. This amount is processed and changed into final products via fixed conversion rates
ρr,z, so the quantity of the refined output z = 1, . . . , Z is
qr,z = ρr,zxr.
The above equality captures the distinguishing feature of the refinery sector, the inverse
Leontief technology described earlier. The total crude intake cannot exceed the refining
capacity, denoted by Kr: xr ≤ Kr. The prices of the refined products pz are determined
by the inverse demand function
pz = pz
(
∑
r
qr,z
)
= pz(Qz), (1.1)
so the prices depend on the total amount of output delivered to the market, Qz. We
mentioned earlier that the demands for different products could be mutually dependent,
i.e. they could be complements or, more likely in case of refinery products, substitutes.
However, this issue is not in the focus of this paper, so in the above formulation we assume
that the demands are independent. Rather, we wish to capture the fact that the products
differ in the value they deliver to the consumers. This can be done by assuming different
price elasticities of the demands.
Now, we can write the total revenue of the refinery as TRr =∑
z qr,zpz(Qz).
Finally, denoting the total costs, being a function of the input and the capacity, by
Cr = Cr(xr, Kr), we obtain the profit function5
πr = TRr − Cr =∑
z
ρr,zxrpz(Qz) − Cr(xr, Kr), r = 1, . . . , R. (1.2)
The objective of each refinery is to choose the optimal crude oil intake xr to maximize
(1.2) subject to the capacity constraint xr ≤ Kr. This maximization problem yields
the optimal input as a function of the remaining R − 1 firms’ inputs. Taken together,
these reaction functions then determine the Cournot equilibrium. Our ultimate goal will
be to find and describe the equilibrium. Next, we proceed with the formulation of the
5Observe that our earlier assumption about crude as a homogeneous input can be slightly relaxed. Infact, having refinery-specific costs, we at least allow for every refinery to use a different type of crude.
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1.3 The Basic Refinery Investment Model
investment model in steps.
1.3.2 Fixed Demand
Let us first consider a simple case, where the demand for each product is assumed to be
fixed at Dz, so the equilibrium price is treated as given, and we may write pz = p∗z. This
can be interpreted as a case of a perfectly competitive market, where all the firms are
price takers. The firms’ optimization problem then boils down to
maxxr≤Kr
πr =∑
z
ρr,zxrp∗z − Cr(xr, Kr)
s.t. Qz ≤ Dz.
We can see that the solution of this problem gives the optimal crude input for every
refinery, which is independent of other refineries’ input unless the demand constraint is
binding.
1.3.3 The General Case
We now turn to the general case, where the price relates to the quantity produced through
the inverse demand function (1.1). We then formulate the firms’ problem as
maxxr≤Kr
πr =∑
z
ρr,zxrpz
(
∑
r
ρr,zxr
)
− Cr(xr, Kr). (1.3)
Solving this problem gives us the input of refinery r as a function of the remaining R− 1
refineries’ input, the reaction function. So we have a system of R equations with R
unknowns, the solution of which yields the Cournot equilibrium.
1.3.4 Upgrade Investment
As discussed in the introductory paragraphs of this section, we would like to find out if
it makes sense for the refineries to invest in more advanced technologies that enable to
extract higher yields of more valuable products from a given amount of crude oil. We can
examine this question by introducing a two-stage game. In the first stage the refineries
(simultaneously) choose the level of investment, which affects their conversion rates but
also their cost functions. Given these levels, an input-choice game is played in the second
stage, as in Section 1.3.2 or as in the general case of Section 1.3.3.
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1.4 Solution of the Model
Let us generalize the notation as follows. The conversion rates now depend on
investment Ir, so that ρr,z = ρr,z(Ir). In particular, after investing Ir, the conversion rate
increases for the higher-valued products, while it decreases for the lower-valued products.
Additionally, the investment affects the costs, so we write Cr = Cr(xr, Kr, Ir). Now, the
2nd-stage profit-maximization problem (of Section 1.3.3) has the form
maxxr≤Kr
πr =∑
z
ρr,z(Ir)xrpz
(
∑
r
ρr,z(Ir)xr
)
− Cr(xr, Kr, Ir).
We can obtain the reaction function of optimal input, which is dependent not only on
other firms’ input, but also on the amount of investment by all the firms. We will then
have xr = x∗r(X¬r, I), where X
¬r = {xi|i 6= r} and I = {I1, . . . , IR}. But since this holds
for every r = 1, . . . , R, we can solve for xr as a function of I alone: xr = x∗r(I).
Knowing the optimal choices of input in the 2nd stage, we proceed backwards to find
the optimal investment strategies. Specifically, we want to find such Ir that solves
maxIr
πr =∑
z
ρr,z(Ir)x∗r(I)pz
(
∑
r
ρr,z(Ir)x∗r(I)
)
− Cr (x∗r(I), Kr, Ir) .
The solution of the above problem produces the optimal investment strategy of firm r
as a function of the investment of the remaining R − 1 firms. Solving this system of R
equations with R unknowns, we obtain the Cournot equilibrium investment strategies.
In the above formulation, the conversion rates and costs are seen as continuously
dependent on investment. However, in the solution of the model we define them for every
particular level of investment, so that finding the Cournot equilibrium involves numerical
comparison of profits for different investment strategies. This is a realistic restriction,
since upgrade investment in the refinery sector is of a discrete nature.
1.4 Solution of the Model
In order to be able to solve the profit-maximization problems formulated in the previous
section, we need to impose further assumptions on the function specifications. A standard
form of the demand function used in the literature is the linear form, which we also use
in this model except for the case of fixed demand. For each product z = 1, . . . Z we have
pz(Qz) = az − bzQz = az − bz
∑
i
ρi,zxi, az > 0, bz > 0.
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1.4 Solution of the Model
As for the cost function, we will use a special logarithmic form:
Cr(xr, Kr) = αr − βr log(Kr − xr), αr > 0, βr > 0.
This function is convenient both intuitively and analytically. It captures the nature of the
production process, where it becomes more difficult to produce as the input approaches
the capacity constraint. Formally, it is a result of the marginal cost going to infinity with
input approaching the capacity:
limxr→Kr
∂Cr(xr, Kr)
∂xr
= limxr→Kr
βr
Kr − xr
= ∞.
Moreover, this property prevents us from obtaining a corner solution of the maximization
problem. Also, the cost function is convex, so the profit is concave, which is a necessary
condition for the existence of maximum. An example of this form of a cost and marginal
cost function is depicted in Figure 1.1. We may now proceed to providing guidance to
CrMCr
xr
0 Kr
Figure 1.1: The cost (full line) and marginal cost (dotted line) functions
solving the model.
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1.4 Solution of the Model
1.4.1 Fixed Demand
Each refinery r = 1, . . . R chooses xr ≤ Kr to maximize the objective function
πr(xr) =∑
z
ρr,zxrp∗z − Cr(xr, Kr) =
∑
z
ρr,zxrp∗z − αr + βr log(Kr − xr)
s.t.
Q1 ≤ D1,...
QZ ≤ DZ .
Solving for xr gives us6
xr = Kr −βr
∑
z ρr,zp∗z(1.4)
Hence, we obtain an interior solution, so that xr < Kr. However, we must assume that
the demand for each product is large enough, so the demand constraints are not binding.7
Thus, we obtain the Cournot equilibrium of optimal crude inputs that are independent
among refineries.
1.4.2 The General Case
Every refinery faces the objective function
πr(xr) =∑
z
ρr,zxrpz(Qz) − αr + βr log(Kr − xr).
Solving the standard profit-maximization problem, we obtain the following expression for
xr :
xr =
(
∑
z
ρr,zaz−∑
z
ρr,zbz
∑
i6=r
ρi,zxi+2Kr
∑
z
ρ2r,zbz
)
4∑
z
ρ2r,zbz
−
√
√
√
√
(
−∑
z
ρr,zaz+∑
z
ρr,zbz
∑
i6=r
ρi,zxi−2Kr
∑
z
ρ2r,zbz
)
2
−8∑
z
ρ2r,zbz
(
Kr
∑
z
ρr,zaz−Kr
∑
z
ρr,zbz
∑
i6=r
ρi,zxi−βr
)
4∑
z
ρ2r,zbz
(1.5)
So we have the optimal input of refinery r as a function of the choices of the remaining
refineries.
6See Appendix for all the derivations.7In fact, accounting for the demand constraints might become a rather complicated issue. Suppose
that the total supply determined from (1.4) exceeds the demand for some product. Yet, that does notnecessarily mean that the equilibrium involves firms reducing their production. It can still be the casethat a loss from the particular product is compensated by a gain from some other product.
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Let us consider an example with R = 2 refineries. Then we have the best response for
refinery 1, given the choice of refinery 2:
x1 =
(
Z∑
z=1
ρ1,zaz−x2
Z∑
z=1
ρ1,zbzρ2,z+2K1
Z∑
z=1
ρ2
1,zbz
)
4Z∑
z=1
ρ2
1,zbz
−
√
√
√
√
(
−Z∑
z=1
ρ1,zaz+x2
Z∑
z=1
ρ1,zbzρ2,z−2K1
Z∑
z=1
ρ2
1,zbz
)
2
−8Z∑
z=1
ρ2
1,zbz
(
K1
Z∑
z=1
ρ1,zaz−x2K1
Z∑
z=1
ρ1,zbzρ2,z−β1
)
4Z∑
z=1
ρ2
1,zbz
,
(1.6)
and we obtain an analogous expression for the best choice of x2 given x1. The Cournot
equilibrium strategies x1 and x2 can then be found by numerical solution.
1.4.3 Upgrade Investment
The 2nd-stage objective (in the general case) is to choose xr ≤ Kr to maximize
πr(xr, Ir) =∑
z
ρr,z(Ir)xrpz
(
∑
r
ρr,z(Ir)xr
)
− Cr(xr, Kr, Ir),
where Ir indicates the level of investment from stage 1. The conversion rates and the
costs are now a function of investment. In particular, we may define
Cr(xr, Kr, Ir) = αr(Ir) − βr(Ir) log(Kr − xr),
so both the parameter of the fixed cost, αr, and the parameter of the marginal cost, βr, is
affected by investment. Moreover, the implicit cost of investment is incorporated in the
change in αr.
Setting R = 2 and solving for the best response of refinery 1, given the choice of
refinery 2 and stage-1 levels of investment by both refineries, we arrive at
x1 =
(
Z∑
z=1
ρ1,z(I1)az−x2
Z∑
z=1
ρ1,z(I1)bzρ2,z(I2)+2K1
Z∑
z=1
ρ2
1,z(I1)bz
)
4Z∑
z=1
ρ2
1,z(I1)bz
−
√
√
√
√
√
√
√
√
√
(
−Z∑
z=1
ρ1,z(I1)az+x2
Z∑
z=1
ρ1,z(I1)bzρ2,z(I2)−2K1
Z∑
z=1
ρ2
1,z(I1)bz
)
2
−8Z∑
z=1
ρ2
1,z(I1)bz
(
K1
Z∑
z=1
ρ1,z(I1)az−x2K1
Z∑
z=1
ρ1,z(I1)bzρ2,z(I2)−β1(I1)
)
4Z∑
z=1
ρ2
1,z(I1)bz
,
(1.7)
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1.5 Extensions of the Basic Model
and we obtain an analogous expression for x2. We can then numerically solve for the
optimal input choices as functions of investment levels, so that x1 = x∗1(I) and x2 = x∗
2(I),
where I = {I1, I2}.Hence, the 1st-stage objective of refinery r = 1, 2 is to choose Ir to maximize
πr(I) =∑
z
ρr,z(Ir)x∗r(I)pz
(
ρ1,z(I1)x∗1(I) + ρ2,z(I2)x
∗2(I)
)
− Cr
(
x∗r(I), Kr, Ir
)
.
Consistent with the discrete nature of the investment, suppose that the variable Ir can
attain two discrete values as follows:
Ir =
1 if refinery r does not invest,
2 if refinery r invests(1.8)
and the conversion rates are defined accordingly, so that ρr,z(2) > ρr,z(1) for some set
of products, while ρr,z(2) ≤ ρr,z(1) for the remainder. We can then find the Cournot
equilibrium by forming a 2 × 2 payoff matrix:
HH
HH
HH
HHI1
I2I2 = 1 I2 = 2
I1 = 1 π1(1, 1), π2(1, 1) π1(1, 2), π2(1, 2)
I1 = 2 π1(2, 1), π2(2, 1) π1(2, 2), π2(2, 2)
(1.9)
1.5 Extensions of the Basic Model
In this section we analyze two important extensions of the basic model. While the first
- the problem of the capacity choice - is straightforward and does not involve significant
mathematical complications, the second - the problem of multiple markets - constitutes
a serious analytical difficulty, and we will therefore exclude it from further consideration.
1.5.1 Capacity Games
A Simple Capacity Game
In the first stage the refineries simultaneously choose capacities. An input-choice game
is then played in the second stage, similar to the one examined in preceding sections.
Solving the maximization problem in (1.3) again yields the reaction functions of inputs,
and - since we now explicitly treat capacities as endogenous variables - of capacities, so
we write xr = x∗r(X¬r, Kr), where X
¬r = {xi|i 6= r}. Hence, we have xr = x∗r(K), where
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K = {K1, . . . , KR}. Returning to the first stage, the objective is to find the capacity to
solve
maxKr
πr =∑
z
ρr,zx∗r(K)pz
(
∑
r
ρr,zx∗r(K)
)
− Cr
(
x∗r(K), Kr
)
. (1.10)
For the case of two refineries and three products and with the function specifications
as before, we obtain the best response as in (1.6). However, now the 1st-stage problem
consists of maximizing profits over capacities. It can be seen that after substituting the
best response into (1.10), the objective function becomes a trivial function of K1 and K2.
Thus, the Cournot equilibrium of capacity-choice strategies can be easily found.
Upgrade Investment with Capacity Choice
A straightforward generalization of the upgrade investment game (see Section 1.3.4) is to
include the problem of the capacity choice. This can be done by introducing investment
in capacity. We assume, as before, that investment Ir affects the conversion rates ρr,z and
the costs. In addition, now investment also affects the capacity, so that Kr = Kr(Ir).
Thus, the problem of capacity choice in the previous section is now translated into the
problem of investment choice. Combining this with the upgrade investment model of
Section 1.3.4, we arrive at the 2nd-stage profit-maximization problem of the form
maxxr≤Kr(Ir)
πr =∑
z
ρr,z(Ir)xrpz
(
∑
r
ρr,z(Ir)xr
)
− Cr
(
xr, Kr(Ir), Ir
)
.
and consequently, the 1st-stage problem of the form
maxIr
πr =∑
z
ρr,z(Ir)x∗r(I)pz
(
∑
r
ρr,z(Ir)x∗r(I)
)
− Cr
(
x∗r(I), Kr(Ir), Ir
)
.
Suppose that demand is linear and the cost is given by
Cr(xr, Kr, Ir) = αr(Ir) − βr(Ir) log(
Kr(Ir) − xr
)
.
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For the case of two refineries, we obtain a similar expression to (1.7) for the best response:
x1 =
(
Z∑
z=1
ρ1,z(I1)az−x2
Z∑
z=1
ρ1,z(I1)bzρ2,z(I2)+2K1(I1)Z∑
z=1
ρ2
1,z(I1)bz
)
4Z∑
z=1
ρ2
1,z(I1)bz
−
√
√
√
√
√
√
√
√
√
(
−Z∑
z=1
ρ1,z(I1)az+x2
Z∑
z=1
ρ1,z(I1)bzρ2,z(I2)−2K1(I1)Z∑
z=1
ρ2
1,z(I1)bz
)
2
−8Z∑
z=1
ρ2
1,z(I1)bz
(
K1(I1)Z∑
z=1
ρ1,z(I1)az−x2K1(I1)Z∑
z=1
ρ1,z(I1)bzρ2,z(I2)−β1(I1)
)
4Z∑
z=1
ρ2
1,z(I1)bz
.
(1.11)
Then, in the simplest case of discrete investment defined by (1.8), the Cournot equilibrium
can be found by forming a 2 × 2 matrix as in (1.9).
1.5.2 Multiple Markets
Up to now, we have implicitly assumed that all the refineries operate on a single geographic
market, so that all the produced amount of output is delivered and sold within this
market.8 However, this is a slightly distorted depiction of reality, since most of the
refineries operate and compete on more than one geographic market. In particular,
interregional and international trade is an important aspect of the competition in the
refinery sector. To properly analyze the interaction of the refineries on multiple markets,
we would need to introduce a spatial oligopoly model and account for transportation
costs. The profit-maximization problem of every refinery would then look similar to the
following:
maxxr,
qmr,1,...qm
r,Z,
m=1,...,M
πr =M
∑
m=1
(
Z∑
z=1
(
qmr,zp
mz (Qm
z ) − TCmr,z(q
mr,z)
)
)
− Cr(xr, Kr), r = 1, . . . , R,
s.t.M
∑
m=1
qmr,z = qr,z = ρr,zxr, z = 1, . . . , Z,
where the upper index m denotes the particular market and TCmr,z(q
mz ) is the cost of
transporting of qr,z amount of product z to market m. Hence, the decision problem of the
refinery consists of choosing the total amount of crude oil input, xr, and (once qr,z = ρr,zxr
of every product is refined) of choosing the amounts of every product to be delivered to
each of the M markets. Clearly, since the spatial oligopoly game involves a choice of
M × Z + 1 variables and strategic interaction on M markets, the analysis of the model
8This is not to be confused with the markets for different products.
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1.6 Comparison with a Single-Product Case
would go well beyond the framework of this paper.9 Thus, in our basic model, we are
forced to stick to the assumption of one geographic market and no external trade. In the
application of the model in Chapter 3, we do return to this issue, yet with a few strict
assumptions.
1.6 Comparison with a Single-Product Case
The legitimate question arises how the multi-product oligopoly model presented earlier
differs from the standard single-product model. We saw that due to the specific technology
that is characterized by a fixed relation between input and output, the firm’s decision
problem reduces to the choice of a single variable, the level of investment in the upgrade
investment game, or the amount of crude oil intake in the capacity utilization game. It is
then natural to think of an analogy with the single-product oligopoly model. In particular,
let the technology be specified by a linear production function
qr = ρrxr.
Hence, the firm’s output is a single homogeneous good, the price of which is again
determined by the inverse demand function
p = p
(
∑
r
qr
)
= p(Q).
The objective of firm r = 1, . . . , R is to maximize
πr = ρrxrp(Q) − Cr(xr, Kr)
over input xr subject to the capacity constraint xr ≤ Kr. In the case of fixed demand,
the above problem can be written as
maxxr≤Kr
πr(xr) = ρrxrp∗ − Cr(xr, Kr)
s.t. Q ≤ D.
9Pompermayer et al. (2002) analyze a refinery oligopoly model that accounts for transportation costsand uses sophisticated linear programming techniques.
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1.7 Qualitative Analysis
Solving for the familiar linear demand and logarithmic cost specifications gives the optimal
input
xr = Kr −βr
ρrp∗.
In the general case we solve the profit-maximization problem of the form
πr(xr) = ρrxrp(Q) − Cr(xr, Kr)
With two refineries the reaction curve is given by
x1 =(ρ1a−x2ρ1bρ2+2K1ρ2
1b)
4ρ2
1b
−√
(−ρ1a+x2ρ1bρ2−2K1ρ2
1b)
2
−8ρ2
1b(K1ρ1a−x2K1ρ1bρ2−β1)
4ρ2
1b
.
We can similarly proceed with the upgrade investment game. Summing up, it is evident
that the multi-product oligopoly model treated in the previous sections is a direct and
straightforward generalization of the simple single-product model, the latter being its
special case. Indeed, by combining the parameters of the multi-product model in a proper
way, we may immediately arrive at the single-product model formulation.
1.7 Qualitative Analysis
Let us now turn our attention back to the basic investment model formulated in Section
1.3. In Section 1.4 we showed that under the assumption of specific functional forms
we are able to find the Cournot equilibrium of the underlying input-choice game, using
expression (1.4) for the case of fixed demand or the reaction functions (1.5) in the general
case. Consequently, defining a discrete relationship between the levels of investment
and the final payoffs enables us to construct a payoff matrix and to find the Cournot
equilibrium of the upgrade investment game. Apparently, the actual solution of the game
involves cumbersome mathematical expressions. Instead of presenting them here, we
rather attempt to provide a simple qualitative analysis of the equilibrium. In particular,
we focus on the case of two refineries that face a decision whether to invest or not in
an upgrading facility. The starting point is the reaction curve derived in the upgrade
investment game, with capacity choice given by (1.11). We simplify the notation by
omitting the dependence on investment and by denoting sums of products of multiple
parameters by a single letter, as follows:
Ar =Z
∑
z=1
ρr,zaz, Br =Z
∑
z=1
ρ2r,zbz, Γ =
Z∑
z=1
ρ1,zρ2,zbz
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Then the reaction function in (1.11) can be rewritten as
x1 =(A1 − Γx2 + 2K1B1) −
√
(−A1 + Γx2 − 2K1B1)2 − 8B1(K1A1 − K1Γx2 − β1)
4B1
.
(1.12)
1.7.1 The Input Choice Game
First, let us examine the optimal input choices and the corresponding profits, given a
particular combination of the investment strategies of the two firms. What we can see
from (1.11) or (1.12) is that the reaction function is downward-sloping10 (as opposed to
the constant inputs in (1.4)), which is a manifesting feature of Cournot oligopoly models.
Figure 1.2 depicts a reaction curve for a particular choice of parameters.
x1
x2
0
Figure 1.2: Reaction curve
Suppose next that the products vary in the value perceived by the consumers in terms
of the price elasticity of demand. In particular, the higher-valued products are less price-
elastic than the lower-valued ones, or, their demand parameters az and bz are greater.11
Then, we may investigate what happens if - all other parameters holding fixed - refinery
1’s yield of some higher-valued product (denote by h) increases in exchange for a decreased
yield of a lower-valued one (denote by l). We can see that this unambiguously increases
parameter A1 and, if the yield of product h is already higher at both refineries, also
parameters B1 and Γ.
A brief qualitative examination of (1.12) reveals that parameter A1 shifts the reaction
function upwards, while parameters B1 and Γ shift and rotate it downwards. It remains
10See Appendix for a formal derivation of this claim.11In case of linear demand pz = az − bzQz, the price elasticity for a current price-quantity pair is given
by ez = −pz/(pz − az) or ez = pz/(bzqz).
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to determine which of these effects prevails. It turns out that for a reasonable choice of
parameters the effect of A1 outweighs the other two parameters only up to some point,
presumably in a region where refinery 1’s yield of product h is lower than that of product
l yet, that is, where parameter B1 can act reversely. After this point, the effect of B1
and Γ prevails. So, initially the reaction curve gradually shifts up and then returns back
down. Figure 1.3 shows an illustration of a reaction function varying with the yield of
the valued product (and of the lower-valued one). As a consequence, transferring some
of the refining yield from a heavy product to a lighter one affects the optimal crude oil
input such that it increases in the beginning and then bends backward.
x1
x2ρ1,h
00 0
Figure 1.3: Reaction curve varying with ρ1,h
The above result is of little surprise. With a higher yield of the valued product it
is optimal for the refinery to attain more crude input to gain the additional profit from
this product. But, at some point, the yield is so high that it can suffice with less input,
so with lower costs. On the other hand, a too high yield of the valued product need
not be beneficial for the refinery. It can happen that even with less input a significant
amount of the product is delivered to the market, which in turn pushes its price down
and hence, the refinery’s profit. Further, we may examine how refinery 2’s optimal choice
and corresponding profit changes with increasing refinery 1’s yield of the valued product.
Since the reaction curve is downward-sloping, the optimal input of refinery 2 will follow
the exact opposite pattern, and so will its profit. Figure 1.4 illustrates the discussed
behavior of optimal input choices and the corresponding profits.
Finally, let us compare the above outcome with the case of fixed demand (perfect
competition). According to (1.4), the optimal input choice increases with a higher yield
of product h, assuming that p∗h > p∗l and that the demand constraints are not binding.
This, however, makes the refinery approach its capacity constraint faster, which in turn
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1.7 Qualitative Analysis
a)
ρ1,h
x1
x2
π
0
b)
ρ1,h
π1
π2
0
Figure 1.4: (a) Optimal input and (b) profit of refinery 1 (full line) and refinery 2 (dotted line) varyingwith ρ1,h
significantly affects the costs. In the end, if the competitive prices are sufficiently high,
there might be overproduction, and high costs close to the capacity constraint may
actually make the firms worse off than in the oligopoly case, consistent with what the
theory of perfect competition would suggest.
All these findings confirm that the possibility of investment in technologies which
enable the transfer of yields is an interesting and relevant issue to study. Even more so if
one needs to account for strategic interaction among refiners and for the discrete character
of investment.
1.7.2 The Upgrade Investment Game
We may now proceed with an analysis of the upgrade investment game. In the simplest
case where the two refineries decide whether to invest or not, we have four possible
combinations of investment strategies, of which we can construct a payoff matrix as
in (1.9). Our aim is to determine which of the four combinations can constitute the
Cournot equilibrium. We have just seen that - all else holding fixed - increasing the
yield of the valued product may be profitable up to some point. However, we omitted
two important factors. First, the other refinery also has an opportunity to invest to
increase its yields of higher-valued products. As noted previously, the reaction curves
are downward-sloping, which means that any action of one refinery induces an opposite
response of the competitor.
Second, the upgrade investment is costly. It is natural to think that the investment is
reflected in the change of the cost parameters as follows. The new cost function will be
flatter, so that the marginal cost approaches infinity slower. This is implicitly taken care
of in case of capacity increase. Then, parameter βr is used to adjust the speed of marginal-
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cost convergence to infinity. However, the fixed cost of increased capacity is higher and
the refinery must also build the new plant, so it incurs the actual cost of investment.
These two elements are transmitted into a change in parameter αr. An illustration of how
investment can affect the cost and marginal cost curves is shown in Figure 1.5.
a)
xr
Cr
0
b)
xr
MCr
0
Figure 1.5: (a) Total cost and (b) marginal cost before (full line) and after (dotted line) investment
Hence, the resulting equilibrium is an outcome of a few diversely acting forces.
Summing up, to evaluate the profitability of the investment strategy, one must take the
following factors into consideration. First, increased yields of valued products raise the
profit only within a certain range. Second, an action by one refinery aimed at increasing
its profits induces a reaction of the competitor, which pushes the profits down. Third,
investment incurs cost.
The equilibrium of the upgrade investment game can be found by a standard method
for finding the Cournot-Nash equilibrium, that is, by indicating the best response of one
player given the strategy of the second player. This means comparing particular cells
of the matrix in (1.9). For instance, given that refinery 2 decides to invest, refinery
1 compares its payoffs π1(1, 2) and π1(2, 2). The profit from investing, π1(2, 2), will be
greater than π1(1, 2) if, first, refinery 1 is not on its backward bending part of the profit
function, that is, ρ1,h(2) is sufficiently low and, second, the associated cost of investment
and the production cost after investment is not too high, that is, α1(2) is sufficiently
low. If the same holds for refinery 2, the Cournot equilibrium will have both refineries
investing.12
Other types of equilibria may arise. If one of the refineries has some technological
advantage, for example, in terms of costs or in terms of capacity, it can be optimal only
12This result is consistent with a rather general finding in the literature that if the firms compete ala Cournot in both investment and production stages, the equilibrium exhibits tendencies toward over-investment by the oligopolists. For reference, see Brander and Spencer (1983) or Reynolds (1986).
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for this refinery to invest. The disadvantaged refinery would either find it too costly
to invest, or it would be discouraged by the fact that already a substantial fraction of
the market is served by the other refinery, so the prices are too low. Another kind of
equilibrium can occur if the cost of investment is too high for both refineries. Then,
obviously, none of them will invest. Finally, a special type of equilibrium arises if the
refineries are technologically symmetric, but the demands are insufficient to accommodate
the increased yields of both refineries. In such a case, only one of the refineries invests in
equilibrium, but which of them actually does cannot be determined by this static analysis.
Rather, it might be the subject of a commitment analysis, where one of the refineries has
a 1st-mover advantage and can commit to investment.
Lastly, let us briefly examine how the upgrade investment game can evolve in the case
of the fixed demand problem in the 2nd-stage. In this case the firms have lower incentive
to invest, since their profit functions tend to bend backward faster, as we mentioned
previously. In fact, a type of the equilibrium where only one refinery invests is then more
likely.
This concludes the first chapter. We are now ready to introduce a major extension of
the model that brings it closer to real-world phenomena, uncertainty. Modeling refineries’
decisions under uncertainty is the focus of the second chapter.
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Chapter 2
Modeling Refinery Investment under
Uncertainty
In the present chapter we introduce the second key feature of the refining industry that
this thesis aims to model, uncertainty. First, a preliminary guide to modeling uncertainty
in the literature is provided. Then, a simple two-period application to our refinery
investment model is presented. A qualitative analysis concludes the chapter.
2.1 Investment under Uncertainty
Investment is defined as an act of incurring expenses now with the prospect of profits
generated at some point in the future. Associated with this act is some degree of
uncertainty over the potential payoff. Specifically, the future reward from the investment
follows a certain probability distribution, the realization of which is not known at the
time when the investment decision is taken. Economists have struggled to develop a
general rule that would be able to evaluate the attractiveness of a particular investment
project and hence to help form optimal investment decisions. The traditional neoclassical
theory presents the net present value (NPV) rule as an appropriate criterion for valuing
investment projects. It assumes calculating the present value of the future cash inflow
generated by the project, from which the present value of the cost necessary to launch
the project is deducted. A positive NPV implies that the project should be undertaken.
However, as pointed out by Dixit and Pindyck (1994), the NPV criterion is based on
one of two crucial assumptions, which are often overlooked. First, the investment project
is considered as fully reversible, or, second, the investor is facing a now-or-never decision.
Yet, these two conditions are rarely met in practice. The cost incurred to initiate the
project is at least partially sunk, so it cannot be fully recovered, if the firm decides later
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to retract the project. This is because the project often involves transaction specificity,
so the purchased assets cannot be sold, or if yes, then only at a discount. Also, unless
strategic considerations such as entry deterrence force a firm to decide quickly, it usually
has some flexibility about the timing of the investment. That is, the firm can postpone
its decision until it acquires more information.
The violation of the two conditions has fostered the development of a new view of
investment, the real options approach. This approach recognizes the opportunity cost of
investment, which stems from the two features - the irreversibility and the ability to delay
investment. This is where the notion of a real option emerges. Similar to financial options,
the firm can choose to invest now (to exercise the option) or to wait until the uncertainty
is at least partly resolved, and thus to take the risk that the value of the project changes.
The actual investment decision is then based on comparing the present value of investing
now with the present value of investing at possible future dates.
Research has shown that the opportunity cost of investment can be large and ignoring
it might lead to erroneous investment decisions. In fact, first attempts to capture
uncertainty together with irreversibility and timing flexibility in investment valuation
models date back to 1970s. Most of the valuation techniques originate in the papers of
Merton (1973) and Black and Scholes (1973), the pioneering works on financial options
pricing. Myers (1977) argues that the optimal exercise of real options can create a
significant corporate value. Since then, the literature has seen many attempts at applying
the real options framework to investment models, among which, some of the prominent
ones being Pindyck’s (1988) analysis of the optimal capacity of a project and Trigeorgis’
(1990) treatment of investment in natural resources. More recently, Imai and Watanabe
(2004) examine investment under uncertainty in a market with the presence of a first-
mover advantage and devise a model which could also be applied to the oil refinery sector.
Cruz and Pommeret (2005) analyze investment with embodied technological progress and
energy price uncertainty.
The standard real options model1 utilizes the tools of stochastic calculus. In particular,
the value of the project is assumed to follow a geometric Brownian motion. Then,
the solution of the model calls for an application of dynamic programming techniques
through the use of the Bellman equation and Ito’s lemma. However, in this paper,
being complicated enough by the multi-product part, we consider only a simple discrete-
time application of the real options framework.2 We will assume that the value of the
investment can change at some discrete time points. This makes the calculation of the
1See Dixit and Pindyck (1994).2Perotti and Kulatilaka (1998) present a discrete-time strategic real option model.
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2.3 Uncertainty in Refinery Investment Models
present values of investing at particular dates considerably simpler.
2.2 Uncertainty in Refinery Investment Models
Since there is a significant lag between investment decisions and their materialization,
the decisions can be viewed as strategic commitments that carry considerable risk. The
future value of the investment project is uncertain due to changing economic conditions.
In our refinery model uncertainty is imposed on the demand side. In particular, we will
assume that at the time when investment decisions are made, the refineries do not know
the future state of the demand when the investment comes into practice.
Demand uncertainty will be modeled as stochastic demand with different states of
the world with commonly known probabilities. Consequently, the refineries base their
decisions on their rational expectations about demand. Consistent with the vast literature
on uncertainty, we will apply the real options approach in our model. To this end,
investment is viewed as, first, irreversible sunk cost and, second, as possible to be delayed.
As opposed to the traditional NPV theory, this approach allows to explicitly account for
the ability to wait and to value the option of delaying investment.
It is important to note that besides demand uncertainty, the refineries can face
other kinds of uncertainty. For instance, at the time when the investment decision is
taken, the refinery has some expectation about the associated cost, but does not know it
precisely. At the time of the materialization of the investment, the cost can turn out to be
larger than previously calculated. Hence, cost uncertainty may emerge as a remarkable
issue. Technically, however, assuming no asymmetry in information,3 modeling cost
uncertainty would not differ much from the treatment of demand uncertainty presented
below. Consequently, only demand uncertainty is considered here.
2.3 The Basic Model with Uncertainty
2.3.1 Setup
In the first chapter we examined the investment behavior of refineries in a static and fully
deterministic setting. We showed that if the refineries have perfect information about
the state of demand for each product, and if they simultaneously choose their investment
plans and crude oil intakes, we can determine the Cournot equilibrium of their investment
3That is, the probability distribution of the costs of every refinery is of common knowledge.
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strategies. In what follows, we develop this model by introducing demand stochasticity
and a multi-period decision algorithm.
Hence, the prices of the refined products are now determined by the inverse demand
function
pz = pz(Qz, εz), z = 1, . . . Z, (2.1)
where εz’s are parameters of random shocks, the (joint) distribution of which is known.
The investment decision is made prior to the realization of this distribution, and therefore,
the refineries’ 2nd- and 1st-stage objective is (in the general case of Section 1.5.1) to
maximize the expected profits:
maxxr≤Kr(Ir)
E[πr] =∑
z
ρr,z(Ir)xrE
[
pz
(
∑
r
ρr,z(Ir)xr, εz
)]
− Cr
(
xr, Kr(Ir), Ir
)
,
and
maxIr
E[πr] =∑
z
ρr,z(Ir)x∗r(I)E
[
pz
(
∑
r
ρr,z(Ir)x∗r(I), εz
)]
− Cr
(
x∗r(I), Kr(Ir), Ir
)
,
respectively.
Now, suppose that the refineries have the opportunity to delay the investment decision
and wait until the realization of the demand shock becomes known. Then, waiting
one period will enable them to resolve demand uncertainty, and they will thus face a
deterministic problem as in Section 1.5.1.
Following the above description, we can construct a simple stepwise decision tree. At
the beginning of the first period, the firm forms its expectation about demand and can
choose either to invest or not to invest. At the end of the period, the random shock is
realized and the firm’s payoff is delivered, based on its decision. Then, at the beginning
of the second period, if the firm decided to invest in the first period, no choices are left
now, but if it decided not to invest (that is, to postpone its decision), the firm - already
knowing the state of demand - can again choose to invest or not to invest. The payoff is
then again realized at the end of the period.
Evidently, a trade-off between deciding to invest now and postponing the decision
until the next period may arise. If the firm decides to invest now, it may gain additional
profit from this investment, provided that the realized demand turns out to be favorable.
On the other hand, if it decides to wait, it may gain from resolving demand uncertainty
and thus making a decision based on actual demand conditions. In particular, should the
demand conditions turn unfavorable, the firm might refrain from investing.
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2.3 The Basic Model with Uncertainty
2.3.2 The Case of One Refinery
Let us examine the above decision process in more detail. We start with the simplest
case, where we assume that only one of the refineries has the investment opportunity.4
Suppose that demand stochasticity is represented by
pz = εzpz(Qz), z = 1, . . . Z, (2.2)
where εz is a binomial random variable defined as
εz =
uz with probability λz,
dz with probability 1 − λz,
where uz > 1 and dz < 1. Thus, in addition to what is determined by inverse demand, the
price of each product z can randomly increase by factor uz or decrease by factor dz. The
realization of this distribution comes at the and of the first period and the firm takes its
period-1 decision based on the expectation of (2.2). The refinery’s choices, together with
the payoffs, are depicted in the tree in Figure 2.1.
Period 1 Period 2
Ibπ1(I) + π1(I)
N
bπ1(N) + π1(I)I
bπ1(N) + π1(N)N
Figure 2.1: The tree of the game with one refinery
The refinery evaluates its options based on the expected payoffs. Denote by φi the
probabilities of all 2Z states of the world
φ = {φi|i = 1, . . . , 2Z} =
{
Z∏
k=1
λθk
k (1 − λk)(1−θk)|{θ1, . . . , θZ} ∈ {0, 1} × · · · × {0, 1}
}
and by πi1(I1) the corresponding realized profits in state i5
πi1(I1) =
∑
z
ρ1,z(I1)x∗1(I1)
(
θzuz + (1 − θz)dz
)
pz (·) − C1
(
x∗1(I1), K1(I1), I1
)
.
4That is, for now, we disregard the competitors from strategic consideration.5Note that since only refinery 1 can invest, the optimal crude input depends now only on the investment
of refinery 1, so that x∗
1= x∗
1(I1).
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Now, for some states of the world investing can yield higher profit, while for other ones
not investing can be more profitable. We denote these subsets of states by G and B,
respectively:
G = {i|πi1(I) > πi
1(N)},B = {i|πi
1(I) ≤ πi1(N)}.
Then, the total expected profit of investing in the first period is
22Z∑
i=1
φiπi1(I) (2.3)
and the total expected profit of delaying investment is
2Z∑
i=1
φiπi1(N) +
∑
i∈G
φiπi1(I) +
∑
i∈B
φiπi1(N) (2.4)
Expression (2.3) says that if the refinery chooses to invest in the first period, it will gain
the profit of investing in both periods, while (2.4) says that if the refinery does not invest
in the first period, it knows that it will invest in the second period if the state of the world
turns out to be good (i ∈ G) and refrain from investing otherwise (i ∈ B). Consequently,
our objective will be to determine which of the two expected profits is greater and hence
to find the optimal investment rule.
2.3.3 The Case of Two Refineries
Let us proceed with a more general case, where two refineries operate on the market and
both have the investment opportunity. Then, in forming the investment rules, strategic
interaction between the two refineries must be taken into consideration. Starting from
the first period, four possible scenarios may arise. First, if both refineries decide to invest,
then no choices are left in period 2. Second and third, if only one refinery invests and
the other one delays the decision, the latter’s choice between investing and not investing
constitutes the problem of period 2. Fourth, if both refineries delay the decision, then in
period 2 a simple investment game between the two refineries is played. The tree of the
whole game with possible strategies and payoffs is depicted in Figure 2.2.
Let again φi be the probabilities of the states of the world, with the corresponding
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2.3 The Basic Model with Uncertainty
Period 1 Period 2
R1
R2
I
R1I
R2 b
(
π1(I, I) + π1(I, I)π2(I, I) + π2(I, I)
)
R1N
R2
b
(
π1(I, N) + π1(I, I)π2(I, N) + π2(I, I)
)
I
b
(
π1(I, N) + π1(I, N)π2(I, N) + π2(I, N)
)
N
R2
N
R1
I
R2I b
(
π1(N, I) + π1(I, I)π2(N, I) + π2(I, I)
)
R2N b
(
π1(N, I) + π1(N, I)π2(N, I) + π2(N, I)
)
R1
N
R2
I
b
(
π1(N, N) + π1(I, I)π2(N, N) + π2(I, I)
)
I
b
(
π1(N, N) + π1(I, N)π2(N, N) + π2(I, N)
)
N
R2N
b
(
π1(N, N) + π1(N, I)π2(N, N) + π2(N, N)
)
I
b
(
π1(N, N) + π1(N, N)π2(N, N) + π2(N, N)
)
N
Figure 2.2: The tree of the game with two refineries
realized profits of refinery r = 1, 2 in state i given investment I = {I1, I2} denoted by
πir(I) =
∑
z
ρr,z(Ir)x∗1(I)
(
θzuz + (1 − θz)dz
)
pz (·) − Cr
(
x∗r(I), Kr(Ir), Ir
)
.
Now, similarly to the one-refinery case, we need to distinguish among the states of
the world that determine optimal strategies of the refineries in the second period. In
particular, we denote by GI1 (BI
1) the subset of states in which investing (not investing)
is more profitable for refinery 1, given that refinery 2 invests, and similarly by GN1 (BN
1 )
the states in which investing (not investing) is more profitable for refinery 1, given that
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2.4 Qualitative Analysis
refinery 2 does not invest:
GI1 = {i|πi
1(I, I) > πi1(N, I)}, GN
1 = {i|πi1(I,N) > πi
1(N,N)},BI
1 = {i|πi1(I, I) ≤ πi
1(N, I)}, BN1 = {i|πi
1(I,N) ≤ πi1(N,N)}.
For refinery 2, the subsets GI2 (BI
2) and GN2 (BN
2 ) are defined analogously.
Then, the expected profits of both refineries for the four possible period-1 scenarios
can be summarized in Table 2.1. Again, our objective is to compare the expected payoffs
and to find the equilibrium of optimal investment rules.
R1\R2 Invest Delay
Invest
22
Z∑
i=1
φiπi1(I, I)
2Z
∑
i=1
φiπi1(I, N) +
∑
i∈GI
2
φiπi1(I, I) +
∑
i∈BI
2
φiπi1(I, N)
22
Z∑
i=1
φiπi2(I, I)
2Z
∑
i=1
φiπi2(I, N) +
∑
i∈GI
2
φiπi2(I, I) +
∑
i∈BI
2
φiπi2(I, N)
Delay
2Z
∑
i=1
φiπi1(N, I) +
∑
i∈GI
1
φiπi1(I, I) +
∑
i∈BI
1
φiπi1(N, I)
2Z
∑
i=1
φiπi1(N, N)
+∑
i∈GI
1∩GI
2
φiπi1(I, I) +
∑
i∈GN
1∩BI
2
φiπi1(I, N)
+∑
i∈BI
1∩GN
2
φiπi1(N, I) +
∑
i∈BN
1∩BN
2
φiπi1(N, N)
2Z
∑
i=1
φiπi2(I, N) +
∑
i∈GI
1
φiπi2(I, I) +
∑
i∈BI
1
φiπi2(N, I)
2Z
∑
i=1
φiπi2(N, N)
+∑
i∈GI
1∩GI
2
φiπi2(I, I) +
∑
i∈GN
1∩BI
2
φiπi2(I, N)
+∑
i∈BI
1∩GN
2
φiπi2(N, I) +
∑
i∈BN
1∩BN
2
φiπi2(N, N)
Table 2.1: The expected payoffs in the two-refinery game
2.4 Qualitative Analysis
Without knowing the exact values of the parameters, it appears rather laborious to
analytically find the solution of the game described above. Instead, we provide a brief
qualitative analysis of the equilibrium and examine how the outcome is affected by varying
different parameters.
2.4.1 One Refinery
In the case of one refinery, the decision problem involves comparing the two expressions
for investing and delaying, (2.3) and (2.4). A trivial case arises if the expected profit of
investing is lower than that of not investing. It is easy to see that the optimal decision
(in the first period) is the same as without the option to delay - the firm will choose not
to invest. Therefore, we consider only the case where the expected profit of investing
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is higher, that is, where the conventional NPV rule would suggest to invest. Clearly, in
order for the value of the option to delay to exist, the refinery must expect that such a
period-2 state exists in which it is profitable to refrain from investing, or, the set B is
nonempty. Then, the decision rule reduces to comparing the expected relative gain in
”bad” states in period 2 to the expected relative loss from not investing in period 1, that
is, to determining the inequality
∑
i∈B
φiπi1(N) −
∑
i∈B
φiπi1(I) >
<
2Z∑
i=1
φiπi1(I) −
2Z∑
i=1
φiπi1(N),
or∑
i∈B
φi
(
πi1(N) − πi
1(I))
><
2Z∑
i=1
φi
(
πi1(I) − πi
1(N))
.
The analysis of how investment affects the realized profit in the current state was
conducted in Section 1.7. It remains to examine the effect of stochasticity. Thus, if the
probabilities of the ”bad” states are sufficiently high and avoiding investment in these
states promises a substantial gain (as a result of a substantial price fall, for instance),
it will be optimal for the refinery to delay its decision and wait until period 2 to see
how the state of the economy evolves. Otherwise the refinery will invest immediately. In
particular, in case of only two states,6 a ”good” (g) and a ”bad” (b) state, the condition
for delaying to be optimal is given by
2φb
φg
>π
g1(I) − π
g1(N)
πb1(N) − πb
1(I).
2.4.2 Two Refineries
To analyze the two-refinery game we start from Table 2.1. Again, a trivial case arises
if for both refineries not investing is (in expectation) more profitable than investing, no
matter what the other refinery does. Then, the equilibrium will have both firms delaying
their decision. Hence, we suppose that this is not the case.
Next, consider the case when refinery 2 decides to invest in the 1st period. Given this,
refinery 1 compares its expected payoff from investing and delaying. But since in the 2nd
period no game between the two refineries will be played, this problem is analogous to
the one treated above. Specifically, refinery 1 would decide to invest immediately if the
probabilities of the ”bad” states are not too high or the relative gain in these states is
low. Conducting the same comparison for refinery 2, we can determine the conditions for
6Or equivalently, two subsets of states that yield the same payoff.
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immediate investment by both refineries to be an equilibrium outcome of the game. In
particular, in case of only two states for each refinery, a ”good” (gI1 and gI
2 , respectively)
and a ”bad” (bI1 and bI
2, respectively) state,7 both refineries will choose to invest if the
following two conditions are met:
2φbI
1
φgI1
<π
gI1
1 (I, I) − πgI1
1 (N, I)
πbI1
1 (N, I) − πbI1
1 (I, I)and 2
φbI2
φgI2
<π
gI2
2 (I, I) − πgI2
2 (I,N)
πbI2
2 (I,N) − πbI2
2 (I, I).
If one of the above conditions is violated, it can be seen that the equilibrium will have
the respective refinery delaying investment and the other one investing.8 If none of the
conditions holds, it is possible that both refineries will delay their decision. However,
deriving an exact condition in such case would be more complex, as one needs to account
for the possibility of a game played in the 2nd period. Intuitively, this type of equilibrium
can arise if both the probabilities of ”bad” states and the corresponding relative gains are
high for both refineries.
This closes our discussion of uncertainty in the refinery investment models. In the last
chapter we apply concepts presented so far to a real-world case study.
7Again, these are to be interpreted rather as subsets of states that yield the same profit for theparticular refinery, given that the other refinery invests.
8This is simply because if, say, refinery 1 preferred investing to delaying given that refinery 2 invests, itwould exhibit the same preference given that refinery 2 delays, since the prospect of refinery 2 refrainingfrom investment even in period 2 cannot ”hurt” its opponent.
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Chapter 3
Application: Refinery Investment in
Hungary and Romania
The aim of the present chapter is to apply the model exposed in the first two chapters to
a specific case study. The players of the upgrade investment game are refineries in two
CEE countries - Hungary and Romania, and their strategic behavior on their common
market is studied. The main task is to adopt the available data on these two countries, so
that the parameters of the model can be calibrated. However, since the model is rather
stylized, to be able to fit the data accurately a number of restrictions will be imposed.
We begin with a brief description of the refining sectors and the demand profiles in the
two countries. Then, the calibration is carried out and the results are discussed.
3.1 The Refining Industry
3.1.1 Hungary
All three of Hungary’s refineries - Duna, Tisza and Zala - are owned by MOL Hungarian
Oil and Gas Plc. (MOL), from 2006 an almost 100%-ly privatized company. However,
the Duna refinery is the only active crude processing refinery, with a distillation capacity
of 164 tb/d.1 The reported capacity utilization rate was rather low until 2001. Then,
MOL closed the distillation capacities at the other two refineries , so the utilization figures
approached the EU average, around 90%. The other two refineries are still used for the
desulfurization of fuels, gasoline blending and bitumen and petrochemicals production.
However, MOL has the option of reactivating the distillation capacity at Tisza in case
demand suddenly increases.
1This and subsequent data come from PFC Energy (2006) report.
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3.1 The Refining Industry
In 2005 the crude intake rose by 10%, bringing volumes back to levels of 2001. Also in
2001, a delayed coker was installed, which enabled the refinery to produce equal product
yields, with 12.5% less crude intake since then.
The Duna refinery is the second largest in the region. Moreover, a majority stake in
a small Slovak site Slovnaft allows MOL to coordinate its commercial strategies. In its
domestic market MOL yields surplus production and exports significant product volumes
to the former Yugoslavia, to Germany and to Austria. The refinery is linked to both
the Druzhba and the Adria crude pipeline systems, but currently makes no use of the
latter. In 2005 it processed 13% domestic crude, the remaining 87% crude was of Russian
origin, Russia’s Lukoil being the company’s main crude supplier. The inland location of
Duna efficiently protects it from product imports, except from Austria’s Schwechat, which
enjoys no technological advantage, though.
Having launched the coker in 2001, Duna is a relatively complex refinery. The site has
reduced the share of heavy fuel oil in its product yield below 3% and increased gasoline
and gasoil yields. Also, MOL invests in upgrading the Duna refinery in order to further
raise gasoline production capacity, but mainly to raise the desulfurization capacity.
The completion of MOL’s privatization gives it the freedom to develop its own long-
term growth strategy, with the prospects of expanding in both upstream and downstream,
in order to remain an influential player in the region. However, increasing dependence
on Russian crude supplies can be risky, since Russian operators may become direct
competitors for regional dominance. Another challenge may come from Austria’s OMV,
which is planning to import additional fuels from its refineries in Romania.
3.1.2 Romania
Romania’s refining sector is one of the longest-established in Europe, and among the
largest and most complex ones in the region. In 2003, the overall refining capacity stood
at 495 tb/d.2 Ten crude-processing refineries operate in Romania, dominated by two
integrated operators, which together control over half of total capacity. The 70 tb/d
Arpechim and the 69 tb/d Petrobrazi refineries are owned by Petrom, while Rompetrol
operates the 100 tb/d Petromidia and the 10 tb/d Vega sites. The rest of the sector
consists of three sites, with a capacity between 56 and 70 tb/d, and very small refineries,
with capacities of less than 10 tb/d.
The capacity utilization rates were very low for a long time, with only about 50% in
2002. Nevertheless, the production remains sufficient to export a substantial fraction
2This and subsequent data come from PFC Energy (2005) report.
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3.2 Demand Profiles
of it. In 2005, production increased significantly after Lukoil reopened its Petrotel
site. However, further capacity reductions can be expected in the coming years, due
to EU accession-driven deregulation. The overall conversion capacity is high relative
to the region, but further investment will be required mainly in the middle distillates
desulfurization capacities.
The refining margins have been among the lowest in Europe, mainly due to an unofficial
price capping system facilitated by the government prior to OMV acquiring its stake in
Petrom. Also, the absence of a connection to the Druzhba pipeline system bars the sector
from acquiring cheap Russian crude. Nearly half of the crude currently processed is of
domestic origin. Nevertheless, the margins are expected to increase after the entrance of
foreign strategic investors - Lukoil and OMV.
Romania’s largest refiner, Petrom, operates the refineries Petrobrazi and Arpechim.
Both source their crude intake partly from imports. In 2003, the Petrobrazi site began
producing EU-compliant gasoline, but was unable to produce the equivalent diesel.
Rather, it has become Romania’s largest LPG producer. In 2002, the fluid catalytic
cracking units were upgraded in both refineries. The Arpechim site is more complex and
is one of Romania’s most advanced refineries in terms of product quality. In 2003, most
of its diesel and gasoline export was EU-compliant.
The privatization of part of Romania’s refining sector promises further improvements.
Importantly, OMV’s acquisition in Romania fits its regional strategies, among which
penetrating the Hungarian market is particularly challenging.
3.2 Demand Profiles
3.2.1 Hungary
The increased use of gas and the decline of the industrial and agricultural sectors caused
Hungarian oil demand to fall through the 1980’s and 1990’s. Accordingly, gasoline demand
declined dramatically in the early 1990’s and continued to fall slowly in the late 1990’s,
only to recover in 2000 to reach an average growth of 1% per annum between 2000 and
2005, driven by increasing car ownership.3 On the other hand, increased road freight
transport and the share of diesel cars triggered a diesel demand growth reaching 9% per
annum between 1995 and 2005. Further, demand for LPG has been growing since 2000,
as was demand for naphta. Finally, demand for fuel oil in power generation rose in the
early 1990’s, but began dropping in the late 1990’s and even more sharply in the early
3Taken from Wood Mackenzie (2006a) report.
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3.2 Demand Profiles
2000s, as a result of switching to gas.
Recently the overall oil demand stabilized and it is anticipated that it grows from 6.8
Mt in 2005 to 8.8 Mt in 2020. Over 75% of that growth will come from the transport
sector, the rest from industry. Thus, gasoline demand is forecast to continue growing
moderately until 2010, when a small drop is predicted due to lower car park growth and
increasing vehicle efficiency. A steady rise in road freight is forecast to further induce
diesel demand to reach 3.4 Mt in 2020, at an annual growth rate of 3%. A modest growth
of LPG demand is forecast to continue, mainly due to a lower level of excise duty on LPG.
Naphta demand is also expected to continue growing to reach 1.4 Mt by 2020. Finally,
fuel oil demand is projected to slowly decline until 2020, driven by the decline of the
heavy industry and by further gasification.
The trends in the supply-demand balance of the refinery products have changed over
the years. Gasoline has been steadily in deficit recently, but surplus is expected to emerge
in the long term, due to presumed increased refinery production in the next years and
a slight decline in demand between 2015 and 2020. Diesel has been in surplus but is
predicted to fall into large deficits until 2010, due to an expected demand growth from
the transport sector. Finally, fuel oil is expected to be balanced slightly in surplus, as a
consequence of declining demand.
3.2.2 Romania
Following the deep restructuring of the economy in the early 1990’s, Romanian oil demand
decreased at an annual rate of 5%.4 However, economic growth in the late 1990’s and the
prospects of the EU accession promising further economic restructuring have boosted the
demand to attain a 2.5% annual growth. This development has also affected the demand
for refined products. Gasoline demand has been increasing since 1995 at a 6% annual
rate. The economic recession in the 1990’s had caused freight transport to decline, and
consequently, to decrease diesel demand by 8% per annum between 1996 and 2000. The
subsequent economic recovery triggered a high growth of freight transport resulting in
5% annual growth in diesel demand. Fuel oil demand declined sharply between 1990 and
2000, as a result of switching to nuclear, coal and gas capacities for power generation.
Total oil demand in Romania is forecast to grow at an average yearly rate of 2% from
11.2 Mt in 2005 to reach 15.2 Mt by 2020. The majority of the growth is predictably
attributed to the transport sector. Gasoline demand is expected to grow strongly at a
5% rate until 2010 due to increased car ownership and economic growth, which is also
4Taken from Wood Mackenzie (2006b) report.
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3.3 Application of the Investment Model
predicted to drive 3% annual growth of diesel demand until 2015. Fuel demand trends
are also anticipated to continue with more replacement by alternative sources.
The trends in the supply-demand balance are rather stabilized. Increased utilization
of the refineries is predicted to outweigh gasoline demand growth, so the gasoline surplus
is expected to rise. Investment in desulphurisation by main refineries will enable them
to boost exports. The diesel surplus is expected to slightly decrease, due to growing
transport demand. Fuel oil has been in a deficit but is expected to shift to a balanced
position by 2015.
3.3 Application of the Investment Model
3.3.1 Assumptions
Our goal is to apply the upgrade investment model to the framework outlined above and
to study the investment behavior of the Hungarian and Romanian refineries. However,
it is clear that we face a few challenges regarding the applicability of our model. Most
importantly, although MOL is a single Hungarian refiner, and thus can be considered
player 1 in the investment game, defining the Romanian player 2 is a little obscure. Ten
refineries operate in Romania, so the supply is rather segmented. We will therefore focus
our attention on the largest Romanian refiner, Petrom, which was recently privatized
by MOL’s major regional competitor, Austria’s OMV. Petrom operates two refineries,
Petrobrazi and Arpechim, accounting for more than half of the total Romanian refinery
production. Hence, a significant restriction to the model must be imposed, the supply of
Romanian fringe is taken as given and thus disregarded from strategic consideration both
by MOL and by Petrom.
Related to this issue is the problem of the model’s single-market requirement.5
Obviously, Hungary and Romania are two markets with MOL and Petrom delivering
products mostly to their domestic markets. Moreover, both refiners export a fraction
of their output and at the same time refined products are imported by some foreign
operators. However, for the purposes of our model it is not a dramatic diversion to treat
Hungary and Romania as a common market with two major competitors, due to their
geographic proximity. Also, similar to the problem of fringe supply, we take the exports
to and imports from out of the region as fixed and thus exclude from refiners’ decision
factors.
A consequence of these limitations is the problematic way the demand is viewed. The
5Recall from Section 1.5.2 what are the difficulties associated with multiple markets.
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demand the refiners are facing is a residual demand, that is, adjusted after accounting
for fringe supply, imports and exports (since we take these as fixed). Accordingly, the
parameters of the demand functions must be adjusted to reflect this issue.
Hence, taking into account the above issues, we construct a refinery duopoly
investment model as follows. We consider the two refineries, MOL and Petrom that
operate on the Hungarian-Romanian market. Their product slate is approximated
by three groups of products, light distillates, medium distillates and residual oil, the
representative products of which being gasoline, diesel and heavy fuel oil, respectively.
Based on the demand profiles, an attractive investment opportunity appears to arise.
In particular, the projected demand shift from fuel oil toward diesel seemingly favors
investing in a hydrocracking unit, which enables to increase diesel yields. This is, however,
a costly investment. Which of the refineries will invest is the core of the game we wish
to model. The final but implicit assumption, thus, is that the real-world problem can
be approximated by our stylized model, mainly in terms of the specifications of the
technological, as well as the demand side.
3.3.2 Calibration
We may now proceed to calibrating the parameters of the basic investment model based
on the available data.6 We use the 2005 data on prices, demands and supplies of refined
products in both countries. Together with the data on price elasticities of demand, which
are adjusted for residual demand, we are able to construct the demand functions. The
way we treat the problematic issues discussed previously is simply by calculating the
current imbalance between supply and demand and attribute this to fringe supply and
international trade. This imbalance is then held fixed. The calibrated demand parameters
are summarized in Table 3.1.7
Product Supply Demand az bz pz = az − bQz
Light 6012 5907 1601 0.17 594Middle 4310 6558 1575 0.23 596Heavy 1929 2093 574 0.17 248
Table 3.1: Calibration of the demand parameters (quantities in kt/year, prices in USD/t)
Further, knowing the refineries’ current conversion rates, capacities and crude inputs,
6The data sources include the PFC Energy (2005, 2006) and the Wood Mackenzie (2006a, 2006b)reports, Petrom 2005 Annual Report, as well as MOL’s private resources.
7The average prices and the aggregate demands and supplies for the two countries were used. Then,using the price elasticities of 0.6, 0.4 and 0.7 (these values are realistic, taking into account the fringesupply and foreign trade) for light, middle and heavy products, respectively, the demand parameters werecalculated. Further, a’s were adjusted to capture the fixed supply-demand imbalance.
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3.3 Application of the Investment Model
we may calibrate the status-quo part of the model, that is, where none of the refiners
invests in upgrading capacity. We do this by equating the crude intake to the expression
in (1.6) and assuming that the players are rational profit maximizers. From (1.6) we then
obtain the only unknown variables that remain to be specified, the parameters of marginal
costs, β1 and β2. Apparently, the parameters of fixed costs (α1, α2) can be taken rather
arbitrarily, since it is only their difference after investment that matters in determining
the equilibrium, as it turns out.
The next step is to specify how the parameters change when the refineries decide to
invest in upgrading capacity. In this we rely on MOL’s expert opinions. In fact, MOL
contemplates investing in a hydrocracking unit, which would increase its diesel yield, as
well as the total refining capacity. In case of Petrom, an increase in the diesel yield is
also possible, but without affecting the total capacity. The cost parameters are then
adjusted as follows. First, before the investment, the fixed costs of both refineries are
approximately at the same level. After the investment, the cost functions become flatter.
For MOL, this is incorporated in the capacity increase, while for Petrom, parameter β2 is
adjusted. Then, the change in parameters α1 and α2 captures the proportional capacity
increase (for MOL) and the cost of investment. In Table 3.2, the parameters for both the
status-quo and the investment part are summarized.
MOL PetromYields Before After Before AfterLight 0.43 0.40 0.46 0.44Middle 0.38 0.44 0.25 0.38Heavy 0.11 0.11 0.18 0.11Capacity 8,100 9,400 8,000 8,000Crude intake 7,100 ? 6,400 ?ProductionLight 3,074 2,938Middle 2,691 1,619Heavy 802 1,126α 2,000,000 2,050,000 3,270,000 3,290,000β 40,883 40,883 183,008 178,000Costs 1,717,587 ? 1,919,811 ?Revenue 3,627,285 ? 2,987,659 ?Profit 1,909,698 ? 1,067,848 ?
Table 3.2: Parameter values before and after the investment (quantities in kt/year)
3.3.3 Results
We now have sufficient data to solve the basic upgrade investment game as in Chapter
1. To fill the question marks in Table 3.2, we construct the matrix of optimal inputs
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3.3 Application of the Investment Model
and the payoff matrix, as in (1.9). These matrices are shown in Table 3.3 and Table 3.4,
respectively. We can immediately find the Cournot equilibrium of investment strategies.
It turns out that the equilibrium has none of the refineries investing. Thus, the prediction
of the static model is that it is not profitable for the refineries to undertake the upgrade
investment, either due to high costs associated with it, or due to relatively low demands.
MOL/Petrom Not invest InvestNot invest 7,100 6,400 6,926 6,041Invest 6,852 6,446 6,594 6,062
Table 3.3: The matrix of optimal inputs
MOL/Petrom Not invest InvestNot invest 1,909,698 1,067,848 1,705,238 1,063,634Invest 1,799,626 1,122,322 1,537,241 1,086,746
Table 3.4: The payoff matrix
Let us proceed with the more realistic setting, the two-period investment game under
uncertainty analyzed in Chapter 2. Based on the demand profiles, we project the future
demand shocks as follows. A moderate growth in gasoline demand and a fair growth in
diesel demand are forecast, while a decline in fuel oil demand is anticipated. Applying to
our framework, we approximate the price shocks by the demand shifts of these products.
In particular, suppose that next year the price of the light and middle distillates may
increase by 5% and 10%, respectively, with a 50% probability, independently of each
other, while the price of the residual fuel may decrease by 10% with a 50% probability.
Assuming that these estimates are common, we wish to find the optimal investment rules
of the two refineries, that is, whether it is profitable to invest now or to postpone the
decision until next year, when the uncertainty is resolved. What we need is to construct
the payoff matrix of the four combinations of the investment strategies, as in Table 2.1.
After some calculations we obtain the payoff matrix shown in Table 3.5.
MOL/Petrom Delay InvestDelay 3,866,434 2,305,107 3,597,967 2,310,037Invest 3,805,237 2,415,661 3,250,397 2,363,942
Table 3.5: The payoff matrix of the two-period game
Hence, we can see that for Petrom the equilibrium investment rule is to invest
immediately, while MOL should delay its decision. It turns out that this result is
robust against various other demand scenarios, provided that the projected diesel demand
increase is sufficiently high. The explanation can be the following. Petrom utilizes the
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3.3 Application of the Investment Model
benefits of investment in all states and investing is a dominant strategy. The potential
gain from the increased diesel yield outweighs the particularly high cost of investment. On
the other hand, MOL, despite being slightly advantaged in terms of costs and capacity,
the advantage in yields is so high that it is already located on the backward bending part
of its profit function (see Section 1.7.2) and thus, rather paradoxically, investing seems
unprofitable for MOL.
We conclude that in the static setting, high investment costs prevents both MOL and
Petrom from investing in upgrading capacity. A prospect of a future demand growth
of diesel and gasoline, though, apparently benefits Petrom and encourages it to invest.
However, this result should be regarded with caution, due to some notable restrictions of
the model’s application.
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Conclusion
The aim of this thesis was to build an oligopoly model which took into account key
characteristics of the oil refining industry. Using game theory tools and elementary
stochastic modeling techniques, the purpose was to capture the following particularities.
First, the output of a refinery is a heterogeneous composite product, so the refinery is
considered as a multi-product firm simultaneously competing in multiple product markets.
Second, due to a relatively high share of each refinery in the regional market, adjusting
the production to meet changing demand involves large-scale investment that carries
significant sunk costs. Third, a high degree of uncertainty over future payoffs is associated
with operating in the refinery market, due to fluctuating market conditions.
In view of recent changes in the demand profiles from heavy products to lighter
products the question arises whether and to what extent it is profitable for the refineries to
undertake the upgrade investment whereby they can increase the refinery yields of lighter,
higher-valued products. In an attempt to answer this question, strategic interaction
among refineries was taken into account, and a two-stage Cournot investment game was
designed. In the first stage, the refineries choose their investment strategies by which
they can build an upgrading capacity, enabling them to reprocess heavy residual output
to obtain higher yields of lighter output. Then, given the firms’ stage-1 choices, a capacity-
utilization game is played, where the refineries choose the optimal crude oil intake, which,
due to the special character of the refining technology, is a single decision variable of the
second stage.
In this static and deterministic setting, the goal was to study the equilibrium behavior
of the refineries and to determine the conditions under which a particular set of investment
strategies is optimal. Due to considerable mathematical complications, the investment
game between only two refineries was focused on, presumably, without loss of economic
insights. Also, consistent with the discrete nature of investment, the game was reduced
to two choices - to invest or not in the upgrading capacity. A simple qualitative analysis
revealed that, as long as one of the refineries enjoyed a sufficient technological advantage,
the equilibrium would have only this refinery investing, otherwise, if the demands were
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high enough and the costs of investment not too high, both refineries would invest in the
equilibrium, consistently with the findings in the literature.
Furthermore, to model uncertainty over future payoffs, stochasticity was imposed on
the demand side. Particularly, in addition to what is determined by the inverse demand,
the price can increase or decrease by some random factor, the probabilities and magnitudes
of these shocks being of common knowledge. Again, to keep the analysis simple, a two-
period game with two refineries was designed. In each period an upgrade investment
game is played, with the difference that the refineries resolve uncertainty only in the
second period, while in the first period they base their decisions on their expectations.
Also, consistent with the real options approach, the investment is irreversible, so once the
refinery decides to invest in the first period, it cannot undo its decision later, should the
demand turn out to be unfavorable.
The purpose of this model was to examine whether it was profitable for the refineries to
invest immediately or to use the option to wait and see the realization of the demand shock,
and, possibly, even to refrain from investing at all, if it turned out to be unprofitable.
Again, a short qualitative analysis revealed that if the probabilities of the unfavorable
states were too high as was the relative gain in these states compared to preemptive
investment, then the refinery would delay its investment decision.
Finally, the applicability of the refinery investment model was illustrated by a case
study. The investment behavior of Hungarian and Romanian refineries was studied.
The projected demand shift from fuel oil toward diesel seemingly favors investing in a
hydrocracking unit, which makes it possible to increase diesel yields. The application
of the investment model attempted to answer the question which of the refineries would
actually invest. Consistently with previous theoretical findings, the results suggest that
although the Hungarian refinery enjoys a slight technological advantage, the cost of the
investment is rather high, resulting in both refineries refraining from investment. However,
when the model with uncertainty was applied and a sufficiently high probability for the
upward shift of the diesel demand was assumed, it was found that it would be profitable
at least for the Romanian refinery to invest immediately, while the Hungarian refinery
would, rather paradoxically, delay the investment decision.
Lastly, it must be noted that throughout the construction of the model, a few
simplifications and diversions from reality were necessary. Most importantly, refined
product markets are regionally segmented by transportation costs, which was, due to
mathematical complication, omitted in the model. Thus, the model deserves further
elaboration and development. However, the major contribution of this thesis is the
combination of two traditional theories in the economic literature - multi-product
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oligopoly and investment under uncertainty - and applying them to analyze strategic
decision making of firms in a particular industry - in the refinery industry. The author
believes that this goal has been fulfilled.
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Appendix
A.1 Derivation of Optimal Input in the Basic Model
For the case of fixed demand the profit-maximization problem yields the first-order
condition (FOC)
0 =∂πr(xr)
∂xr
=∑
z
ρr,zp∗z −
∂Cr(xr, Kr)
∂xr
=∑
z
ρr,zp∗z −
βr
Kr − xr
.
Solving for xr gives
xr = Kr −βr
∑
z ρr,zp∗z.
In the general case we have the FOC:
0 =∂πr(xr)
∂xr
=∑
z
ρr,zpz(Qz) +∑
z
ρr,zxr
∂pz(Qz)
∂xr
− ∂Cr(xr, Kr)
∂xr
=∑
z
ρr,z
(
az − bz
∑
i
ρi,zxi
)
− xr
∑
z
ρ2r,zbz −
βr
Kr − xr
=∑
z
ρr,zaz −∑
z
ρr,zbz
∑
i6=r
ρi,zxi − 2xr
∑
z
ρ2r,zbz −
βr
Kr − xr
Rearranging gives
(
2∑
z
ρ2r,zbz
)
x2r
+
(
−∑
z
ρr,zaz +∑
z
ρr,zbz
∑
i6=r
ρi,zxi − 2Kr
∑
z
ρ2r,zbz
)
xr
+
(
Kr
∑
z
ρr,zaz − Kr
∑
z
ρr,zbz
∑
i6=r
ρi,zxi − βr
)
= 0
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Solving for xr and applying the capacity constraint xr < Kr we have
xr =
(
∑
z
ρr,zaz−∑
z
ρr,zbz
∑
i6=r
ρi,zxi+2Kr
∑
z
ρ2r,zbz
)
4∑
z
ρ2r,zbz
−
√
√
√
√
(
−∑
z
ρr,zaz+∑
z
ρr,zbz
∑
i6=r
ρi,zxi−2Kr
∑
z
ρ2r,zbz
)
2
−8∑
z
ρ2r,zbz
(
Kr
∑
z
ρr,zaz−Kr
∑
z
ρr,zbz
∑
i6=r
ρi,zxi−βr
)
4∑
z
ρ2r,zbz
A.2 Derivation of the Slope of the Reaction Function
To show that the reaction curve is downward-sloping, let us differentiate (1.12) with
respect to x2. We obtain
−Γ + A1Γ−x2Γ2−2ΓK1B1√A2
1−2A1x2Γ−4A1K1B1+x2
2Γ2+4x2ΓK1B1+4K2
1B2
1+8B1β1
4B1
.
Denoting the expression in the numerator by N and solving it for x2 we obtain
x2 =2A1NΓ + A1N
2 − 4K1B1NΓ − 2K1B1N2 ± 2
√2(Γ + N)
√
−N(2Γ + N)B1β1
ΓN(2Γ + N).
We can see that for N > 0 the expression under the root sign is negative, since all the
other parameters are positive. For N = 0, the expression does not make sense. It follows
that there exists no real x2 for which the reaction is upward-sloping or constant. Hence,
it must be downward-sloping.
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